Gravity Lagrangian Research Articles

Averaged Lagrangians and MacCallum-Taub's limit in macroscopic gravity

The derivation of MacCallum-Taub's averaged high-frequency Lagrangian [1] is analysed with special attention paid to the assumptions made along the derivation. It is shown that averaged high-frequency Lagrangians of the same form as MacCallum-Taub's Lagrangian can be derived by applying the Brill-Hartle and macroscopic gravity averaging schemes. A procedure for the derivation of a Lagrangian of macroscopic gravity (an averaged Hilbert action) is proposed and its high-frequency limit (namely, its high-frequency perturbation expansion up to the second order terms in perturbations, which is referred to as MacCallum-Taub's limit) is calculated. There is disagreement [2] in the expressions for MacCallum-Taub's averaged high-frequency Lagrangian and the high-frequency limit of the macroscopic gravity Lagrangian. Possible reasons for such disagreement are analysed. The origin of the difference is shown to consist in using the propagation equation for perturbations, i.e. the linearized Ricci tensor vanishes, during the derivation (averaging) carried out in [1]. A new derivation of an averaged high-frequency Lagrangian without assuming the propagation equation to hold and by taking into account the proper correlation functions is given. The newly derived expression is shown to coincide with MacCallum-Taub's limit of the macroscopic gravity Lagrangian, which resolves the disagreement.

Erratum: Inflatonless inflation

Whichever could be the real theory of gravitation, the corresponding low-energy effective lagrangian will probably contain higher derivative terms. In this work we study the general conditions on those terms in order to produce enough inflation to solve some of the problems of the standard Friedmann-Robertson-Walker cosmology in absence of any inflaton field. We apply our results to some particular scenarios where higher derivative terms appear in the effective lagrangian for gravity like those coming from graviton (two)-loops or integrating out ordinary matter (like the one present in the Standard Model).

Inflatonless inflation.

Whichever turns out to be the real theory of gravitation, the corresponding low-energy effective Lagrangian will probably contain higher derivative terms. In this work we study the general conditions on those terms in order to produce enough inflation to solve some of the problems of the standard Friedmann-Robertson-Walker cosmology in the absence of any inflaton field. We apply our results to some particular scenarios where higher derivative terms appear in the effective Lagrangian for gravity, such as those coming from graviton (two) loops or integrating out ordinary mater (such as the one present in the standard model).

CONFORMALLY INVARIANT LAGRANGIANS IN METRIC-AFFINE AND RIEMANN-CARTAN SPACES

Various types of generalized conformal transformations in non-Riemannian manifolds and corresponding transformation properties of the geometrical objects are reviewed. Possible structures of conformally invariant gravity and matter Lagrangians in arbitrary space dimensions are discussed. New electrovac exact solutions of R+R2+Q2 gravity with conformally flat Bertotti-Robinson metric and nonzero torsion have been found.

Some consequences of the effective low-energy lagrangian for gravity

We cconsider the minimal low-energy action for gravity up to six derivatives which is renormalizable at the two-loop level modulo higher derivative corrections. Then we study the classical solutionscorresponding to the Schwarzchild and the Robertson-Walker metrics. In the first case we find a singularity close to the gravitational radius and in the second case we find inflationary de Sitter solutions in absence of any matter or cosmological constant.

Ghost- and tachyon-free gauge-invariant, Poincar�, affine and projective Lagrangians

We prove the existence of new classes of ghost- and tachyon- free gauge-invariant gravity Lagrangians for the spontaneously broken Poincare, affine, and projective groups, respectively. To do so the formalism of spin-projection semioperators which generalizes the one of the ordinary spin-projection operators has been developed.

Classical hair for Kerr-Newman black holes in string gravity

We compute the external field strength hair present for black holes in string gravity. We retain all couplings in the effective lagrangian for string gravity through first order in α′, the string tension. We work in general black hole backgrounds, both rotating and charged, and compute both dilaton and axion hair to that order, thus expanding and systematizing previous treatments.

From Chern-Simons to Gauss-Bonnet

The relation between SO(1, 2 p − 1) Chern-Simons forms and SO(1, 2 p − 2) Gauss-Bonnet forms in 2 p − 1 dimensions is discussed in detail based on recent results by Chamseddine. This approach singles out a special linear combination of Gauss-Bonnet terms as a lagrangian for gravity in odd dimensions. We show that in eleven dimensions this theory naturally admits spontaneous compactification over the four-dimensional Minkowski space and in some sense even distinguishes four dimensions. Similar results are obtained in any odd dimension. We also find a certain degeneracy of the field equations which might cause a problem for such a “topological gauge theory of gravity”.

Ashtekar's variables, the θ-term, and the cosmological constant

Capovilla, Jacobson and Dell have given a lagrangian for pure gravity in which only an SO(3) connection and a Lagrange multiplier appears. We show that the introduction of a cosmological constant leads to a non-polynomial action, which we determine to tenth order in the curvature. We point out that, in Ashtekar's hamiltonian formulation, the cosmological constant can be introduced by a transformation that is formally similar to the transformation which leads to a θ-term.

Two-dimensional gravity with dynamical torsion and strings

A two-dimensional theory of gravity with dynamical metric and torsion that was proposed in the context of string theory is considered. We analyse the equations of motion following from a Lagrangian quadratic in curvature and torsion, L = R 2 + T 2 + λ. The equations of motion are reduced to a system of equations for two scalar fields which in a particular case yields the Liouville equation. We have found and analysed stationary solutions with nontrivial torsion. It is proved that equations for extremals defining trajectories of pointlike spinless particles are completely integrable. A singularity of stationary space-time with nontrivial torsion has a repulsive character contrary to that of a black hole, and no particle can penetrate through it. Modification of the string theory due to the addition of the gravity Lagrangian is discussed.

Very large angular scales and very high energy physics

The early production of gravitational waves in different inflationary scenarios is most simply analyzed in a conformally transformed metric where it is possible to write and discuss a simple expression for their amplitude at the post-inflationary horizon reenter. Three special cases are considered here: generalized gravity lagrangians L( R), scalar-tensor, and multidimensional (or Kaluza-Klein) theories. In the latter case any upper limit on Δ T/ T may set a lower bound to the compactification ratio of the inner radius b( t)/ b 0 during the de Sitter phase.

Lagrangians for conformal gauge gravity and conformal simple supergravity

The connection, curvature, and Lagrangian for a conformal gauge gravity are obtained. A set of generators of the conformal simple supergroup is given, the commutation and anticommutation relations for the superalgebra are calculated, and a Lagrangian of the simple supergravity is established.

A lagrangian for topological gravity and its BRST quantization

A gauge invariant lagrangian for topological gravity is presented. The BRST action associated to this lagrangian has features similar to the one proposed by Witten. It is found that topological gravity can be consistently BRST quantized without coupling gravity to matter fields, while preserving conformal invatriance.

Hamiltonian analysis of extra gauge symmetries in an R+T2 theory of gravity.

We study the Hamiltonian structure of a class of gravity Lagrangians which are linear in the curvature and quadratic in the torsion, in the case when extra gauge symmetries are present in the theory. The corresponding symmetry generators are found by using Dirac's general method for systems with constraints, and the dynamical meaning of extra gauge symmetries is clarified.

Dimensionally continued Euler forms: Kaluza-Klein cosmology and dimensional reduction

The most general gravity Lagrangian in more than four dimensions is considered which leads to field equations with at most second derivatives of the metric. It consists of a series of dimensionally continued Euler forms and allows spontaneous compactification. The field equations are elaborated for the usual Kaluza-Klein cosmology ansatz and solved in the special case where the extra dimensions form a sphere with constant radius. The dimensional reduction of the theory to four dimensions is discussed as well.

Symmetric solutions to the Gauss-Bonnet extended Einstein equations

Low-energy limits of string theories suggest that gravity lagrangians should include quadratic and higheorder curvature terms, in the form of dimensionally continued Gauss-Bonnet densities. In an arbitrary number of dimensions, we consider the static, spherically symmetric solutions to the lowest order Gauss-Bonnet extended Einstein equations. We also find isotropic, homogeneous cosmological solutions with an ideal fluid source.

Supersymmetric formulation of gravity theories with nontrivial Lorentz connection.

Using Breitenlohner's algebra, we study the supersymmetric extension of the 9-parameter gravity Lagrangian of Sezgin and van Nieuwenhuizen, which has the vierbein and Lorentz connection as independent fields. The requirements that the theory be unitary and invariant under local supersymmetry transformations are so strict that the only feasible supersymmetric extension is ordinary supergravity. The same holds for a more general 17-parameter Lagrangian considered by Neville. We discuss the close relationship between our results and the structure of the supersymmetry transformations of Breitenlohner's potentials.

Scaling behavior of semiclassical gravity

Using the idea of metric scaling we examine the scaling behavior of the stress tensor of a scalar quantum field in curved space-time. The renormalization of the stress tensor results in a departure from naive scaling. We view the process of renormalizing the stress tensor as being equivalent to renormalizing the coupling constants in the Lagrangian for gravity (with terms quadratic in the curvature included). Thus the scaling of the stress tensor is interpreted as a nonnaive scaling of these coupling constants. In particular, we find that the cosmological constant and the gravitational constant approach UV fixed points. The constants associated with the terms which are quadratic in the curvature logarithmically diverge. This suggests that quantum gravity is asymptotically scale invariant.

Extra “gauge” symmetries in R + T 2 gravitational theories

We investigate a class of gravity lagrangians which are linear in curvature and quadratic in torsion terms. It is found that for special cases these theories are invariant with respect to local transformations beyond general coordinate and Lorentz transformations. These additional invariances are worked out explicitly andtheir relevance is discussed. The “gravity with extra gauge symmetry” by Bars and MacDowell turns out to be just one example.

Hamiltonian structure of the theory of gravity withR+T 2 type of Lagrangian

The Hamiltonian structure of a class of gravity Lagrangians which are linear in the scalar curvature and quadratic in the torsion is studied by using Dirac’s general method for systems with constraints. It is found that the Lorentz gauge potentials can be eliminated from the theory by using constraint equations, without affecting the Dirac brackets of the remaining variables.